Abstract
Various connexive FDE-based modal logics are studied. Some of these logics contain a conditional that is both connexive and strict, thereby highlighting that strictness and connexivity of a conditional do not exclude each other. In particular, the connexive modal logics cBK\(^{-}\), cKN4, scBK\(^{-}\), scKN4, cMBL, and scMBL are introduced semantically by means of classes of Kripke models. The logics cBK\(^{-}\) and cKN4 are connexive variants of the FDE-based modal logics BK\(^{-}\) and KN4 with a weak and a strong implication, respectively. The system cMBL is a connexive variant of the modal bilattice logic MBL. The latter is a modal extension of Arieli and Avron’s logic of logical bilattices and is characterized by a class of Kripke models with a four-valued accessibility relation. In the systems scBK\(^{-}\), scKN4, and scMBL, the conditional is both connexive and strict. Sound and complete tableau calculi for all these logics are presented and used to show that the entailment relations of the systems under consideration are decidable for finite premise set. Moreover, the logics \(\mathbf {cBK}^-\) and \(\mathbf {cMBL}\) are shown to be algebraizable. The algebraizability of \(\mathbf {cMBL}\) is derived from proving \(\mathbf {cMBL}\) to be definitionally equivalent to \(\mathbf {MBL}\). All connexive modal logics studied in this paper are decidable, paraconsistent, and inconsistent but non-trivial logics.
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Notes
- 1.
- 2.
When Iacona introduces the strict conditional view as a view of indicative conditionals, this seems to presuppose that we are dealing with natural language conditionals. However, one can, and Iacona does, also consider formal languages containing a conditional that is meant to represent a natural language indicative conditional. Also, note that the claim that indicative and subjunctive conditionals are distinct is contentious, see Priest (2018). Iacona (2019, p. 2) explains that his focus on indicative conditionals “is not intended to suggest that counterfactuals differ in some important respect. On the contrary, most of what will be said about conditionals can be extended, mutatis mutandis, to counterfactuals.” We will not take a stance on this issue here. Moreover, when we are just interested in defining the strictness and the connexivity of a conditional, we need not define the notion of a conditional but may take it as given. The notion of an implication is often introduced by requiring that it is a binary connective satisfying the Deduction Theorem, see, for example, Avron et al. (2018), Wansing and Odintsov (2016). Note also that we will use the terms “conditional” and ‘implication” as synonymous.
- 3.
Usually, it also required that a connexive implication, \(\rightarrow \), is non-symmetric, i.e., that \((A \rightarrow B) \rightarrow (B \rightarrow A)\) fails to be a theorem.
- 4.
Note that also for Richard Routley (Routley et al. 1982), the failure of conjunctive simplification is characteristic of connexive logic. For Routley, connexive logic and relevance logic more or less coincide. If one shares the containment view of valid implication according to which in a valid implication \(A \rightarrow B\), the content of B must be included in the content of A, then disjunctive addition fails, and if connexive logic is motivated by the idea of negation as cancelation, then conjunctive simplification cannot hold in full generality, in particular, \((A\wedge \sim A) \supset A\) and \((A\wedge \sim A) \supset \sim A\) fail to be valid (cf. also Wansing and Skurt 2018).
- 5.
Since the metalanguage is classical, all classical equivalences hold in the metalanguage.
- 6.
As usual, \(\Box ^0 B=B\) and \(\Box ^{n+1} B=\Box \Box ^n B\).
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Acknowledgements
The work reported in this paper has been carried out as part of the research project FDE-based modal logics, supported by the Deutsche Forschungsgemeinschaft, DFG, grant WA 936/13-1, and the Russian Foundation for Basic Research, RFBR, grant No. 18-501-12019. We gratefully acknowledge this support.
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Odintsov, S., Skurt, D., Wansing, H. (2021). Connexive Variants of Modal Logics Over FDE. In: Arieli, O., Zamansky, A. (eds) Arnon Avron on Semantics and Proof Theory of Non-Classical Logics. Outstanding Contributions to Logic, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-030-71258-7_13
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